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The main image looks something like a circle with one point representing both negative and positive infinity. I find this image confusing. I picture the real projective line as a real line with two added points, Inf and -Inf. An even better picture in my mind I get is just the real number line and two added dots. Is this image misleading or not? I don't have any reference texts beside me to cite at the moment, sorry. 137.124.161.17 ( talk) 00:19, 26 January 2016 (UTC)
There are articles on projectively extended real line and projective line that overlap severely with this one. The first could probably be eliminated and seems to be a WP:FORK. Possibly merge all three. 73.89.25.252 ( talk) 23:18, 15 June 2020 (UTC)
The group GL2(R) acts on the left on the space of column vectors R2 by matrix multiplication. In this way, each matrix A defines a function fA from R2 to R2. If B is another matrix, then because holds for all column vectors . Thus matrix multiplication is compatible with composition, when one uses the left action on column vectors. The same then follows for the induced left action of PGL2(R) on P1(R).
Indeed, it is this left action on P1(R), with acting as , that one finds in standard references such as
(Some of these references are over C, but the principle is the same over every field.)
If anyone disagrees with all these authors, can you please explain why and list the published references you are using to support your claim? Thank you, Ebony Jackson ( talk) 04:59, 27 December 2020 (UTC)
Thank you for your response! I have heard that several decades ago, there was a pocket of mathematicians in the U.K. who started writing functions to the right of their arguments, so that to apply the composition f∘g one would apply f first and g second, as you mention. But it was a losing battle, and I think that essentially all of them abandoned that notation. Even J. W. P. Hirschfeld, it seems, now writes his transformations on projective space using matrices acting on the left: see p. 4 of his 2008 book Algebraic curves over a finite field. And Rafael Artzy seems to have passed away in 2006, so one can hardly call him a practicing geometer.
Writing functions to the left of their arguments is the standard in virtually all of mathematics, not just calculus and complex analysis, and this includes almost all branches of geometry as well. The vast majority of users of the projective line will be more familiar with this standard convention. If you still feel it is worth mentioning that there is a convention in which transformations are written on the right, this could be done in a note with a citation to a book that uses this convention, but I feel that it would be wrong to present this convention in the main text as if it were standard in the 21st century.
By the way, by PSL(R) do you mean PSL2(R) = SL2(R)/{±1}? That group is not the full group of linear automorphisms of P1(R). Ebony Jackson ( talk) 04:50, 30 December 2020 (UTC)
![]() | This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
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The main image looks something like a circle with one point representing both negative and positive infinity. I find this image confusing. I picture the real projective line as a real line with two added points, Inf and -Inf. An even better picture in my mind I get is just the real number line and two added dots. Is this image misleading or not? I don't have any reference texts beside me to cite at the moment, sorry. 137.124.161.17 ( talk) 00:19, 26 January 2016 (UTC)
There are articles on projectively extended real line and projective line that overlap severely with this one. The first could probably be eliminated and seems to be a WP:FORK. Possibly merge all three. 73.89.25.252 ( talk) 23:18, 15 June 2020 (UTC)
The group GL2(R) acts on the left on the space of column vectors R2 by matrix multiplication. In this way, each matrix A defines a function fA from R2 to R2. If B is another matrix, then because holds for all column vectors . Thus matrix multiplication is compatible with composition, when one uses the left action on column vectors. The same then follows for the induced left action of PGL2(R) on P1(R).
Indeed, it is this left action on P1(R), with acting as , that one finds in standard references such as
(Some of these references are over C, but the principle is the same over every field.)
If anyone disagrees with all these authors, can you please explain why and list the published references you are using to support your claim? Thank you, Ebony Jackson ( talk) 04:59, 27 December 2020 (UTC)
Thank you for your response! I have heard that several decades ago, there was a pocket of mathematicians in the U.K. who started writing functions to the right of their arguments, so that to apply the composition f∘g one would apply f first and g second, as you mention. But it was a losing battle, and I think that essentially all of them abandoned that notation. Even J. W. P. Hirschfeld, it seems, now writes his transformations on projective space using matrices acting on the left: see p. 4 of his 2008 book Algebraic curves over a finite field. And Rafael Artzy seems to have passed away in 2006, so one can hardly call him a practicing geometer.
Writing functions to the left of their arguments is the standard in virtually all of mathematics, not just calculus and complex analysis, and this includes almost all branches of geometry as well. The vast majority of users of the projective line will be more familiar with this standard convention. If you still feel it is worth mentioning that there is a convention in which transformations are written on the right, this could be done in a note with a citation to a book that uses this convention, but I feel that it would be wrong to present this convention in the main text as if it were standard in the 21st century.
By the way, by PSL(R) do you mean PSL2(R) = SL2(R)/{±1}? That group is not the full group of linear automorphisms of P1(R). Ebony Jackson ( talk) 04:50, 30 December 2020 (UTC)